REPORT MAS-R0112 AUGUST 2001

Centrum voor Wiskunde en Informatica

MAS Modelling, Analysis and Simulation

Modelling, Analysis and Simulation Riemann-problem and level-set approaches for two-fluid flow computations I. Linearized Godunov scheme B. Koren, M.R. Lewis, E.H. van Brummelen, B. van Leer REPORT MAS-R0112 AUGUST 2001

CWI is the National Research Institute for Mathematics and Computer Science. It is sponsored by the Netherlands Organization for Scientific Research (NWO). CWI is a founding member of ERCIM, the European Research Consortium for Informatics and Mathematics. CWI's research has a theme-oriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms. Probability, Networks and Algorithms (PNA) Software Engineering (SEN) Modelling, Analysis and Simulation (MAS) Information Systems (INS)

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Riemann-Problem and Level-Set Approaches for Two-Fluid Flow Computations I. Linearized Godunov Scheme B. Koren, M.R. Lewis and E.H. van Brummelen

CWI P.O. Box 94079, 1090 GB Amsterdam, The Netherlands

B. van Leer

The University of Michigan, Department of Aerospace Engineering Ann Arbor, MI 48109-2140, USA

ABSTRACT A nite-volume method is presented for the computation of compressible ows of two immiscible uids at very dierent densities. The novel ingredient in the method is a two- uid linearized Godunov scheme, allowing for ux computations in case of dierent uids (e.g., water and air) left and right of a cell face. A level-set technique is employed to distinguish between the two uids. The level-set equation is incorporated into the system of hyperbolic conservation laws. 2000 Mathematics Subject Classi cation: 65M60, 76N99, 76T10 Keywords and Phrases: free surfaces, compressible liquid-gas ows, interface capturing, Godunov scheme, level-set method. Note: This research was supported by the Dutch Technology Foundation STW (applied science division of NWO and the technology programme of the Ministry of Economic Aairs) and by the Maritime Research Institute Netherlands. The work was carried out under CWI-project MAS2.1 \Computational Fluid Dynamics".

1. Introduction

1.1 Application area: water-air ows The present paper is directed towards the computation of two- uid ows consisting of water and air. Premises are that the water and air do not mix and that vaporization and condensation phenomena do not occur. Furthermore, the surface water waves that are of practical relevance are assumed to be of suciently large scale to safely neglect surface tension. Despite these simpli cations, the application area is still vast: computational ship hydrodynamics, the combined computation of wind elds and sea currents, etc. Various computational challenges exist in accurately and eciently computing water-air

ows, in particular their interfaces. 1.2 Treatment of free-surface ows 1.2.1 Basic grid techniques To compute a free surface on a grid, basically two techniques exist: (i) moving or tting (Lagrangean) techniques and (ii) xed-grid (Eulerian) techniques. In the rst, the free surface forms (part of) the grid boundary (Figure 1a); the latter is t to the free surface. In the second, the free surface is in general not aligned with the grid (Figure 1b). Because in the moving-grid approach the grid is attached to the free boundary, its free surface is crisp, but it is not necessarily accurate; its location and topology may be poorly resolved. A moving-grid technique is not well-suited for the computation of bifurcating free surfaces. In practice, it may even be unsuited for free surfaces that are non-bifurcating, but just strongly distorted. Fixed-grid techniques do not suer from such drawbacks, but here the front may be diused. Mixed Lagrangean-Eulerian methods,

2

which try to combine the best of the above two basic techniques, may be constructed. An example of such a hybrid approach would be to overlay a xed grid with a narrow moving grid attached to the free surface (Figure 1c). We proceed by considering moving- and xed-grid techniques in more detail.

a. Moving

b. Fixed

c. Hybrid

Figure 1: Types of grid techniques for computing free-surface ows. 1.2.2 Moving-grid techniques A common approach in moving-grid techniques is to describe the free surface as a height function, say as h(x; t) in the 2D unsteady case (Figure 2), or as h(x; y; t) in 3D. Some references to free-surface height methods are [1, 2, 3]. It is common use to do the calculation of the free-surface height separately from that of the bulk ow, a reason being that in case of a single-valued height function, the free-surface- ow problem is one dimension lower than the bulk ow problem. (Methods that compute the height function and the bulk ow simultaneously do exist though, see, e.g., [4].) y

h

x

Figure 2: Free-surface height h. The number of conditions to be imposed on the free surface is one more than on xed boundaries, it is equal to the usual number for the bulk ow (two conditions for the 2D Navier-Stokes equations, e.g.) plus one extra for the height function. For free-surface- ow problems as occurring in ship hydrodynamics, the simplest free-surface conditions express that the mass transport through and the stresses at the free surface are zero. Imposing these conditions in a physically correct way is of paramount importance for a good resolution of the free surface and is not trivial. One subtlety in case of, e.g., steady free-surface water waves is that these are known to satisfy dispersion relations [5], which uniquely relate the lengths of the free-surface waves under the action of gravity to the Froude number. Ideally, in the discrete case, these relations should be satis ed. Another point of attention is the intersection of the free surface and a no-slip boundary, an example of this is the water line at a ship's hull. Here, the (minor) diculty is that in the unsteady case the free-surface and the no-slip boundary conditions do not match.

1. Introduction

3

Because during the free surface's motion the grid may strongly distort, discretization of the ow equations in the computational domain is more appropriate than in the physical domain. A consequence is that because of the more intricate equations (due to the metric terms), it is harder to build in physics at a low discretization level. For steady, free-surface- ow problems, a common solution approach (when using the free-surfaceheight method) is the following two-step process: 1. Solve the steady bulk- ow equations (the full Navier-Stokes equations, e.g.) with the free surface frozen and with { consequently { one free-surface condition not imposed. 2. On the basis of the previously obtained bulk- ow solution, correct the free surface such that it (better) satis es the free-surface condition not imposed in the rst step. If not yet converged for both bulk ow and free-surface ow, return to step 1. A pitfall in making this solution process ecient is to make both steps ecient, ignoring their interaction. This holds in particular for step 2; introducing large free-surface updates may hamper or even ruin the bulk- ow problem's convergence to a steady state. For that reason, the rule in steady free-surface ow computations is to carefully march the unsteady free-surface and bulk- ow equations to a steady state. However, because of the persistent unsteady free-surface- ow phenomena, this evolutionary approach is not ecient at all. In [6], still following the above two-step process, an ecient method is proposed for solving the steady free-surface ow (step 2) whilst keeping the induced perturbations in the bulk ow (step 1) small. The key lies in the implementation of the free-surface conditions: in [6], a so-called quasi-free-surface condition is proposed. For smooth, singly-connected free surfaces, this method appears to work perfectly well [7, 8]. However, when steepening the water waves, as expected, its performance deteriorates and nally breaks down. In case of unsteady free-surface ows, the technique of alternatingly updating the bulk ow and the free surface may be followed as well. However, it is expected that this subcycling approach is only rst-order accurate in time, independent of the separate accuracies of the time integrators for bulk ow and free-surface

ow. (Evidence for this, in the application area of uid-structure interactions, is given in [9].) Fixed-grid methods hold out the promise of resolving a much larger class of free-surface ows and { also { of not suering from a time accuracy barrier in the unsteady case. In the next section, we consider some pros and cons of xed-grid techniques. 1.2.3 Fixed-grid techniques In the Eulerian approach, since many years, some well-proven techniques exist for computing ows with free boundaries. A classical method is the Marker-and-Cell (MAC) method [10]. In it, one of the two uids is seeded with massless particles that go with the ow. A grid cell without any particle is de ned to be a cell fully lled with the other uid (possibly void) and the free surface(s) may be de ned as the set(s) of cell faces separating the cells with particles from the cells without, or { more accurately { as the tight contour(s) wrapped around the particles such that no cell without particles is closed in (Figure 3a). Flow bifurcations cause no diculties to the MAC method. A de ciency though is that no clear distinction can be made between physical and numerical cavitation (Figure 3a: empty cell at the bottom of the uid). To avoid this uncertainty, the rule is to seed the uid with as many particles as possible. Doing so, particularly in 3D, the MAC method may become very expensive. A more ecient xed-grid method is the Volume-of-Fluid (VOF) method [11]. In it, per grid cell, in addition to the standard uid- ow quantities such as velocity and pressure, a scalar quantity is introduced, which represents the fraction of that cell, that is lled with one of the two uids: the VOF fraction (Figure 3b). The fraction is transported by means of the advection equation. The location of the free surface can be de ned in a similar way as in the MAC method, viz. as the set of cell faces separating the cells with the VOF fraction larger than zero from the cells with the VOF fraction equal to zero, or { more accurately { at the subcell level. The latter requires intricate ux algorithms, particularly in 3D. A principal drawback of the VOF method is that the VOF fraction is non-smooth and { hence { hard to accurately resolve in precisely the region of interest: at the free surface.

4

A natural x is to replace the VOF fraction by a smooth scalar function, which represents, e.g., the distance to the free surface. Doing so, the free surface is simply de ned as the zero-distance iso-surface. This is known as a level-set method (Figure 3c). A text book on level-set methods is [12], a classical journal paper is [13]. Older work in which the technique is already found, though not yet under the name of level-set technique, is that by Markstein on ame propagation. See, e.g., [14], in which a ame front is represented by a higher-dimensional, dierentiable function that is advected by the ow. Since last decade, level-set methods enjoy many publications, a list related to CFD only is [15]{[30]. When computing the ow of a single uid with free surface, i.e., a material-void interface, a minor problem of level-set methods is that no velocity eld is de ned in the void region. Hence, to guarantee smoothness of the level-set function, a proper arti cial velocity eld in the void region needs to be de ned. This diculty may also be seen as a good opportunity though to improve the free boundary's resolution without being inhibited by physics. E.g., in the void region an arti cial velocity might be chosen which counteracts the eects of numerical diusion by anti-advection [23, 31]. Keeping the level-set function smooth forms a point of attention. During the computation, the function may need to be regularized. In this reinitialization step, care needs to be taken that the free-surface location is preserved. 0

0

0

0

0

0

0

0

0

0 .1 .1 0

0

0

0 .3 .8 1 .7 0

0

0 .3 .8 1 .8 .4 0

0 .1 .7 1 0 .6 1

? a. MAC

0

0

1 .2 0

1 .9 0

0

0

-.3 -.3 -.2 -.2 -.2 -.2 -.2 -.2 -.3 -.2 -.2 -.1 -.1 0 -.1 -.1 -.2 -.2 -.1 0 -.2 -.1 0

0

0

0 -.1

0 .1 0

0 -.1

0

-.1 -.1 0 .1 0 -.1 -.1 -.1

0

-.1 0 .1 .1 0 -.1 -.1 -.1

.8 1

1

1 .8 0 .7 0

0

1

1

1

1 .8 0

.1 .1 .2 .1 0 -.1 -.1 -.1

0

0

0 .1 .1 0 -.1 0 -.1

1

1

1

1

1 .6 .1 0

.2 .2 .2 .1 0

1

1

1

1

1

1 .9 .7

.3 .3 .2 .1 .1 .1 0

1

1

1

1

1

1

1

1

.4 .3 .3 .2 .2 .1 .1 .1

1

1

1

1

1

1

1

1

.4 .4 .3 .3 .2 .2 .2 .2

b. VOF

0 -.1 -.1 0

c. Level-Set

Figure 3: Known types of xed-grid techniques for computing free-surface ows. Using an MAC, a VOF or a level-set method, free-surface boundary conditions can be imposed at all cell faces separating the two uids or { alternatively { at the subcell level. With a level-set method, the latter can be done more accurately than with the VOF method, but is still intricate, particularly in 3D. In [32], the approach of imposing free-surface conditions in the interior of the computational domain is called tracking. In some sense, imposing conditions on a free surface is a contradiction in terms. The free surface is genuinely free when it is captured (i.e., when no conditions at all are imposed on it). In capturing methods, in principle, the free surface is a two- uid interface (with one

uid possibly virtual). The challenge of capturing methods is to choose or devise a physically correct two- uid ux formula to be applied at the low discrete level of cell faces. A speci c, well-known problem of capturing (not of tracking and tting) techniques is that large solution errors may occur near the interface (the so-called `pressure-oscillation problem'). In literature, several remedies against this have already been proposed. In a forthcoming paper we will also address this problem. As mentioned, both tracking and capturing can be combined with the MAC, the VOF as well as the level-set method. In the present paper, we consider the combination of a capturing scheme and a levelset technique. A novel capturing scheme suitable for two- uid ows, in particular liquid-gas (water-air) problems, is proposed. As compared to the surface-height method, with the MAC, VOF and level-set

1. Introduction

5

methods, computational overhead has been introduced by the representation of the n-dimensional free surface as an (n + 1)-dimensional set of particles (MAC) or function (VOF and level-set). When computing a single- uid- ow problem with free surface by a capturing technique, even more overhead is introduced by the computation of the ow of the virtual second uid. The overhead is expected to be counterbalanced to an amply sucient degree by the advantages of capturing: no remeshing, no limitation with respect to free-surface topologies, no free-surface boundary conditions to be imposed (the dispersion relations in case of water waves under the action of gravity, e.g., will be satis ed automatically), etc. 1.3 Point of departure The present work is directed towards two- uid- ow computations on the basis of the Navier-Stokes equations, excluding phenomena such as surface tension, mixing of the uids, and condensation and vaporization. Diusion and gravity are still planned for future work, here the homogeneous Euler equations are considered. In [32], Kelecy & Pletcher consider capturing for incompressible two- uid- ow equations, which are balanced through the zero-divergence relation for the velocity eld. In a capturing technique for the ow of two uids with dierent densities, density is an unknown even in the incompressible case. Given this fact, a compressible approach will not necessarily be much more computing-intensive. The foreseen applications are homentropic, yielding { for perfect gases and liquids { equations of state of the compact form p = p(). Hence, to balance the equations, no energy equation needs to be invoked. Use of the compressible uid- ow equations has the following advantages over the incompressible:  The unpleasant property of weak coupling between continuity equation and momentum equations (through velocity only) is avoided. Advantage can be taken of numerical techniques with built-in physics, which have been developed already for gas-dynamics applications. E.g., at each cell face, a local 1D Riemann problem can be considered (Godunov approach). With this, it should be possible to resolve in a physically correct way the two- uid interface (represented by the contact discontinuity) over a single cell face or cell only. In such a compressible approach, there will be no need for, e.g., staggering or { in case of unsteady ow simulation { pseudo-time stepping.  In principle, the compressible approach is always more correct. For very low-Mach number ows, it may still be preferable when, e.g., large characteristic lengths play a role. The time needed by pressure disturbances to travel along a ship's hull, e.g., may be signi cant. (Denoting the characteristic length and time by l and  , respectively, and the ow speed and speed of sound by u and c, with the for ship hydrodynamics realistic values l = 100 m, u + c  c  1000 m/sec, it follows  = l=(u + c) = 0:1 sec, which may not be negligible in unsteady ow simulations.) In low-Mach-number hydraulics problems such as the simulation of the impact of storm surges on dikes, compressibility is even mandatory. (Air enclosed by water wave and dike will behave as an airbag; compressibility is essential in the dynamics of these ow problems.) For the known diculties of low-Mach number ows (poor solvability and reduced accuracy) many xes have been proposed, see, e.g., [33]{[41], and [42], p. 578 for more recent work. 1.4 Outline of paper In the present paper, the emphasis still lies on the development of a liquid-gas Godunov-type scheme, in combination with a level-set technique, i.e., on ows with a single spatial dimension only. The contents of the paper is the following. In Section 2, the continuous ow model is given (conservation laws, equation of state and level-set equation). In Section 3, the space discretization of the equations is presented (the Riemann problem and corresponding Godunov-type scheme, at both interior and boundary cell faces). In Appendix A, we show that this scheme is a linearized, two- uid Godunov scheme.

6 2. Flow model

2.1 Conservation equations The two uids do not mix. Denoting the densities of the two uids by 1 and 2 , this implies that if 1 > 0 then 2 = 0 and { vice versa { if 2 > 0 then 1 = 0. (See Figure 4 for a 1D illustration with xfs the location of the free surface.) ρ1

0

ρ2

x fs

x

a. Fluid 1

0

x fs

x

b. Fluid 2

Figure 4: Density distributions in 1D ow of two immiscible uids. The immiscibility also implies that u1 (xfs ) = u2 (xfs ) = u(xfs ); the free surface is a contact discontinuity. The separate masses of the two uids need to be conserved. In 1D, this means for a control volume :

Z d1 dt dx + (1 u)@ right , (1 u)@ left = 0;

(2.1a)

Z d2 dt dx + (2 u)@ right , (2 u)@ left = 0:

(2.1b)





R

For the control volume we may next de ne the bulk density   (R1 +dx2 )dx , which for suciently small can be approximated by  = 1 VV11 ++V22V2 , where V1 and V2 are the sizes of the subvolumes of

that are lled with uid 1 and uid 2, respectively. (So, it has been assumed that 1 and 2 are constant over V1 and V2 , respectively.) Introducing the volume fraction  of uid 1,  = V1V+1V2 , we can write

 = 1 + (1 , )2 ;  2 [0; 1]:

(2.2)

An alternative for (2.1) is then

Z d dt dx + (u)@ right , (u)@ left = 0;

(2.3)

plus a still to be lled in equation for the location(s) xfs of the interface(s) in space and time. The latter equation determines  = (x; t). Looking ahead at the dierences between both formulations in a numerical implementation, we already note that with the bulk-density formulation, in a nitevolume discretization, total mass of the uid will be conserved, but not necessarily the masses of the two separate uids. In case (x; t) is not exactly resolved, the two separate masses are not conserved. As opposed to that, with formulation (2.1) in a nite-volume discretization, the masses of the separate

uids are always exactly conserved. Hence, when using formulation (2.3), an accurate resolution of the interface location(s) is of paramount importance. As far as momentum is concerned, the bulk density is a more practical quantity than the densities of the two separate uids, because { as opposed to mass { the momenta of the two uids do not need

7

2. Flow model

to be conserved separately. Only the total amount does, the two uids can exchange momentum. Because surface tension is not considered, it also holds p1 (xfs ) = p2 (xfs ) = p(xfs ). So, for the momentum equation we can write Z d(u) dx + (u2 + p) , (u2 + p) = 0: (2.4)

dt

@ right

@ left

To describe two- uid ows, we opt for the bulk-density description, i.e., equations (2.2){(2.4). The system of equations is not yet balanced. There are six unknowns: 1 ; 2 ; ; p; u and . The equation for the location of the interface (determining ) still has to be chosen. For this, we follow a level-set approach, to be discussed in the next section. For the remaining unknowns 1 and 2 , equations of state 1 (p) and 2 (p) are chosen in Section 2.3. 2.2 level-set equation As mentioned, to conserve the masses of the separate uids as accurately as possible when using the bulk-density formulation, it is essential to resolve the free-surface locations as accurately as possible. For that purpose, the level-set approach is (in principle) better suited than the VOF approach, because of its better smoothness properties. Good smoothness of the level-set function is rst taken care of in the level-set function's initialization. A common approach is to initialize the level-set function as the signed distance to the initial free surface, with the distance positive in { say { uid 1 and negative in

uid 2. The choice for the signed-distance function may not be attractive though. In case of, e.g., a 1D ow problem with two interfaces (Figure 5), the level-set function de ned as the signed-distance fluid 1

fluid 2 ( x fs ) 1

fluid 1

x

( x fs ) 2

Figure 5: 1D two- uid ow with two interfaces. function would look as depicted in Figure 6a, i.e., perfectly smooth at both free surfaces, but with a non-dierentiability in between. Denoting the level-set function by , in formula, the function in Figure 6a reads (x) = min (x , (xfs )1 ; (xfs )2 , x) : (2.5a) The level-set function does not need to be the signed-distance function. An alternative for it, which is uniformly smooth but non-zero at the free surface is sketched in Figure 6b. In formula, here, one may think of, e.g., (x) = e,(x,(xfs )1 )2 (x,(xfs )2 )2 , 1: (2.5b) In [23], numerical experiments are done with level-set functions similar to the latter. For our present applications, the level-set function does not need to be uniformly smooth, only at the interfaces it must be. It is preferred to be linear there. Therefore, in the present paper we do take the signed-distance function as the initial level-set function. The level-set function is advected by, in 1D:

@ + u @ = 0; (2.6) @t @x with u the local velocity. Combined with bulk-mass conservation equation (2.3), quasi-linear equation (2.6) may be written in the conservative control-volume form Z d() dx + (u) , (u) = 0:

dt

@ right

@ left

(2.7)

8 φ

0

φ

(xfs )1

(xfs )2

a. Signed-distance function

x

0

(xfs )1

(xfs )2

x

b. Continuously differentiable function

Figure 6: Possible initial level-set functions for 1D two- uid ow with two interfaces. Conservation of  is not important, there is no conservation law for it. The form (2.7) is simply practical because it is consistent with the system (2.3){(2.4), it can be directly added to it. As soon as the level-set function becomes insuciently smooth during its advection (assuming there is a smoothness criterion available), it must be regularized. Crucial in this reinitialization is that the free-surface location(s) at that speci c moment are preserved as accurately as possible. (If this is not taken care of, the reinitialization can be even counterproductive.) The advantages of level-set methods over particularly VOF methods are:  Level-set functions are smooth at physical discontinuities and { hence { can be advected in a numerically accurate way precisely there. As opposed to that, the less smooth VOF function may easily smear out or become non-monotonous during its advection, thus deteriorating the resolution of the free surface.  The level-set equation can be directly embedded into the system of uid- ow equations and discretized collectively and consistently with these. E.g., it can be included into the Godunovtype scheme, which is what we will do. Related to this, there is no principal diculty in extending a 1D level-set technique to multi-D. 2.3 Equations of state In our water-air computations, for both uids, elegant use can be made of a single equation of state, Tait's [43, 44]:   p + Bp1 =  ; (2.8) (1 + B )p1 1 where the subscript 1 refers to some reference state. For water, it holds (at sea level conditions):

= 7, B = 3000, 1 = 1000 kg=m3, and for air: = 57 , B = 0, 1 = 1 kg=m3. With (2.8), both the water and air density, to be denoted from now on by w (p) and a (p), are convex functions of pressure. Likewise, the corresponding bulk density (; p) = ()w (p) + (1 , ())a (p); () 2 [0; 1] (2.9) is. The physical consequences of this overall convexity are that neither locally very low speeds of sound (much lower than in pure water or pure air), nor entropy-condition-satisfying expansion shocks can occur. These two anomalous phenomena are typical for ows with mixed convex-concave equations of state, ows with, e.g., condensation or vaporization [45], and cannot occur in the immiscible two- uid

ows considered here. To give some more evidence of this, consider the speed of sound of the bulk @p . Using (2.9), we can write 1 =  + 1, , with c2 = dw ,1 and c2 = da ,1 . So,

uid: c2 = @ w a c2 c2w c2a dp dp 2 c2 w a c2 = c2 +c(1 , )c2 : a

w

(2.10)

9

3. Discretization

Assuming that for any given p, ca < cw , from (2.10) it is seen that ca  c  cw for all  2 [0; 1]. For clarity, we also give a graphical illustration. In two-phase ows with condensation or vaporization, the pressure-density diagram may look as sketched in Figure 7a, i.e., as a mixed convex-concave curve dp , in the condensation/vaporization zone. As with extremely small values of the speed of sound, d opposed to that, in the case of two immiscible uids, a family of purely convex curves exists, curves that become increasingly steeper for increasing  (Figure 7b). So, for any p and for all values of  2 (0; 1) it holds ca < c < cw . A slight inconvenience of the nonlinear equation of state (2.8) in combination with (2.9) is that the calculation of p for known  and  ( 6= 0 and  6= 1) needs to be done iteratively. p

α=1/4 α=1/2 α=3/4

p

liquid gas

condensation vaporization

liquid

gas ρ

0

a. For two-phase flows

0

α=0

α=1

ρ

b. For immiscible two-fluid flows

Figure 7: Pressure-density diagrams. 3. Discretization

3.1 Finite volumes Summarizing, for a (suciently small) control volume , the system of equations considered reads

0  1 0 u 1 Z dq @ A @ 2 A dt dx + (f (q))@ right , (f (q))@ left = 0; q = u ; f (q) = u + p ; 



 = ()w (p) + (1 , ())a (p);  p + Bap1  1a w (p) = (1 + B )p (w )1 ; a (p) = (1 + B )p (a )1 ; w 1 a 1 and with () the fraction of the size of over which   0.

 p + Bw p1  1w

u

(3.1a)

(3.1b)

The natural discretization for (3.1a) is a nite-volume technique. We consider cell-centered nite volumes with, for simplicity, constant mesh size h. This choice directly allows us to work out the discretization of (). Consider nite volume i and its left and right neighbors (Figure 8) and de ne the level-set values at the cell faces @ i, 12 and @ i+ 21 as

i, 21 = 21 (i,1 + i ); i+ 12 = 12 (i + i+1 ):

(3.2)

Then, for i  0, we propose the following expression for i :

i, 21  0; i+ 21  0 : i = 1;

(3.3a)

10 h

h

h

Ω i-1

Ωi

Ωi+1

δΩ i- 1

x

δΩ i+ 1

2

2

Figure 8: Finite volume i and its two neighbors.

!

i, 12 < 0; i+ 21  0 : i = 12  ,i 1 + 1 ; i i, 2

(3.3b)

!

i, 21  0; i+ 21 < 0 : i = 12 1 +  ,i 1 ; i i+ 2

(3.3c)

!

i, 12 < 0; i+ 21 < 0 : i = 12  ,i 1 +  ,i 1 : i i i, 2 i+ 2

(3.3d)

The four possibilities in (3.3) are illustrated in Figure 9. So, in determining i, 21 and i+ 12 , as well as x( = 0), use is made of piecewise linear interpolation of . The linear interpolation is exact as long as the level-set function is the signed-distance function. (As soon as this distance property is lost, the exact conservation of the separate water and air masses is lost.) For i < 0, similar expressions can be written. (Using this similarity, the system of expressions for i can be coded compactly.) φ

0

1

i- 12 i-

1 2

i

a.

i+

1 2

i+ 2 i

b.

i+

1 2

i-

1 2

1

1

i- 2

i+ 2

i

i

c.

d.

Figure 9: Four possible combinations of signs of i, 12 and i+ 21 , i  0. 3.2 Riemann-problem approach For the control-volume formulation, we need a formula for the ux vector across a cell face. The formula must have built-in physics for accurately capturing , the free surface.
For that purpose, trivial, term-by-term ux formulae such as, e.g., (f (q))i+ 21 = f 21 (qi + qi+1 ) are less appropriate than formulae derived from the Riemann problem. Besides a good capturing of the free surface, a Riemann problem approach is expected to yield robustness and a good boundary-condition treatment. For ship hydrodynamics applications, a physically proper discretization of convection terms may be as relevant as for aircraft and spacecraft aerodynamics. The convection phenomena in ship hydrodynamics are less rich (no supersonic speeds), but the Reynolds numbers are generally higher (up to O(109 )). The exact solution of the 1D Riemann problem on each cell face, Godunov's approach [46], requires the exact computation of the cell-face state. For the current equations, this implies the use of a numerical

11

3. Discretization

root nder. We avoid this by considering an approximate Riemann solver: Osher's [47], to start with. Denoting the left and right cell-face state by q0 and q1 , and the ux formula by F (q0 ; q1 ), Osher's approximate Riemann solver may be written as

F (q0 ; q1 ) = f (q0 ) + ,

Z q1 df , q0

dq dq;

(3.4)

df . The eigenvalues of the present Jacobian Jacobian dq with dfdq the negative q @p eigenvalue part ofqthe @p are: 1 = u , @ , 2 = u , 3 = u + @p @ . ( @ does not occur in the wave speeds.) For the right eigenvectors we refer to [48], presented there for the 3D situation. The Riemann-invariant relations describing the two intermediate states q 13 and q 23 along the wave path in state space (Figure 10) are

u 31 +

Z  13 1 s @p

 @ d = u0 +

Z 0 1 s @p

 @ d;

(3.5a)

 31 = 0 ;

(3.5b)

u 31 = u 23 = u 21 ;

(3.6a)

p 31 = p 32 = p 21 ;

(3.6b)

u 32 ,

Z  23 1 s @p

 @ d = u1 ,

Z 1 1 s @p

 @ d;

(3.7a)

 32 = 1 :

(3.7b) λ2 =u q1/3 λ1 =u-

δp δρ q0

q2/3 λ3 =u+

δp δρ

q1

Figure 10: Wave path in state space for immiscible two- uid ow. The level-set function  can change along the subpath corresponding with the eigenvalue 2 , i.e., as it should be, across the contact discontinuity. It is invariant along the outer subpaths, i.e., along both subpaths the distance to the free surface is constant. Through the known Riemann invariants u and p at the contact discontinuity, the kinematic and dynamic free-surface boundary conditions (zero mass transport and stresses) are satis ed. The integrals in (3.5a) and (3.7a) can be written out explicitly for the equations given in (3.1b). However, when a free surface is captured, i.e., when  13 and  32 dier in sign, explicit calculation of u 21 and p 21 is hampered by nonlinearity, the aforementioned drawback of Osher's scheme. (A transcendental equation needs to be solved.) Of course, u 12 and p 12 can then be determined numerically. Given foreseen future applications in which u 21 and p 21 need to be dierentiated with respect to q0 and q1 , we do not do so and propose another approximate Riemann solver in the next section.

12

3.3 Two- uid, linearized Godunov scheme Since  is constant along the two outer subpaths of the wave path in Figure 10, along both subpaths the bulk density can only vary due to pressure changes. Because the ows to be considered are uniformlyqlow-subsonic, largeqdensity changes will not occur there and { consequently { the inR  23 1 @p d can be linearized by good approximation around  and  , R tegrals  31 1 @p 0 1 @ d and  @ respectively. Linearization of (3.5a) and (3.7a) yields

u 21 = u0 , ( 13 , 0 ) c0 ;

(3.8a)

u 21 = u1 + ( 23 , 1 ) c1 :

(3.8b)

0 1

Likewise, p 21 can be linearized around 0 and 1 :

p 21 = p0 + ( 31 , 0 )c20 ;

(3.9a)

p 21 = p1 + ( 32 , 1 )c21 :

(3.9b)

Elimination of  13 , 0 and  32 , 1 from (3.8) and (3.9) yields

i.e.,

p 12 , p0 u 21 , u0 = ,C0 ; C0  0 c0 ;

(3.10a)

p 21 , p1 u 21 , u1 = C1 ; C1  1 c1 ;

(3.10b)

 u1  0 2 @ p1 = 2

C0 u0 +C1 u1 +(p0 ,p1 ) C0 +C1 C1 p0 +C0 p1 +C0 C1 (u0 ,u1 ) C0 +C1

1 A;

(3.11a)

with for the density and level-set function in the two intermediate points:

 1  p 1 ,p0 ! 22  + 0 3 c0 ; 1 =

(3.11b)

 2  p 1 ,p1 ! 22  + 1 3 c1 = : 2 

(3.11c)

3

3

0 1

Excluding all supersonic possibilities from the matrix in Figure 11, which shows all possible combinations of eigenvalue signs along the wave path, (note the consequent improvement in eciency!) the two- uid, linearized Osher scheme reads then:

0 1 u1 3 2 B 2 B 1  u F (q0 ; q1 ) = f (q 31 ) = @ 3 12 + p 21  31 u 21  13

1 CC ; A

if u 12  0;

(3.12a)

13

3. Discretization : sonic point + + +

+ + +

+ + +

+ + +

u1/2 > c1/3 -

-

-

- + +

+

-

+

- + +

+

+ - + +

- + +

0 < u1/2 < c1/3 -

-

-

- - +

+

+

- - +

-

+

+ - - +

- - +

-c 2/3 < u1/2 < 0 -

-

-

+

+

- - -

- - -

-

+

+ - - -

- - -

u1/2 < -c 2/3 -

-

-

+

-

+

+

+

u 0 < c0

u 0 < c0

u 0 > c0

u 0 > c0

u 1 < -c1

u 1 > -c1

u 1 < -c1

u 1 > -c1

Figure 11: All possible eigenvalue-sign combinations along wave path from Figure 10, crossed out: all supersonic speeds.

0 2 u1 3 2 B 2 F (q0 ; q1 ) = f (q 32 ) = B @  23 u 12 + p 21  32 u 21  23

1 CC ; A

if u 12  0:

(3.12b)

Note that the real nonlinear ux functions f (q 31 ) and f (q 23 ) are applied, and not F (q0 ; q1 ) = f (q0 ) + (q0 ) if u 1  0 and F (q ; q ) = f (q )+(q 2 , q ) df (q1 ) if u 1  0. There is no need for the latter (q 31 , q0 ) dfdq 0 1 1 1 dq 2 3 2 linearized formulae. On the contrary, as opposed to (3.11){(3.12), they may give rise to an erroneous, (q0 ) and f (q ) + (q 2 , q ) df (q1 ) may ambiguous ux at u 12 = 0 (steady free surface); f (q0 ) + (q 31 , q0) dfdq 1 dq 1 3 be dierent for u 21 = 0. For the single- uid case, (3.12) reduces to

0  12 u 12 F (q0 ; q1 ) = @ 2

1 A ;  12 = (p 21 );  21 u 12 + p 12

(3.13)

with u 21 and p 21 still given by (3.11a). The latter scheme is known, see Section 2.3 and Appendix A in [49], and also Section 9.3 in [50]. It is the single- uid, linearized Godunov scheme. In Appendix A of the present paper, through a (partial) derivation of the (exact) Godunov scheme, it is shown that the present two- uid, linearized Osher scheme (3.10){(3.11) is in fact the two- uid, linearized Godunov scheme. For clarity, in the following, we only use the latter name for scheme (3.11){(3.12).

14

3.4 Boundary-condition treatment The boundary of the computational domain is formed by cell faces. The uxes across the boundary faces can be computed with (3.12) as well. Denoting the state at the boundary by qb , in case of a left boundary q0 = qb and in case of a right q1 = qb . For (subsonic) in ow, in 1D, two of the three components of qb must be imposed, reducing the wave path of the two- uid, linearized Godunov scheme (Figure 12). In case of (subsonic) out ow, a single component of qb must be imposed, leading to the reduced paths given in Figure 13. qb

q2/3

q1/3

q1

qb

q0

a. Left boundary

b. Right boundary

Figure 12: Reduced wave paths for in ow boundary. qb

qb

q1 a. Left boundary

q0 b. Right boundary

Figure 13: Reduced wave paths for out ow boundary. With (3.4), for left in ow and left out ow it holds, as it should be, F (qb ; q1 ) = f (qb ) and for both right cases, correctly as well, F (q0 ; qb ) = f (q0 ) + f (qb ) , f (q0 ) = f (qb ). We work out the in ow and out ow boundary, and the non-permeable boundary as the limit case. For all three it holds, for boundary at the left:

pb , p1 ub , u1 = C1 ;

(3.14a)

pb , p0 ub , u0 = ,C0 :

(3.14b)

and at the right:

3.4.1 In ow boundary The two boundary conditions to be imposed here cannot be ub and pb simultaneously; when ub is imposed, pb follows from (3.14). Vice versa, when pb is imposed, ub follows from (3.14). Hence, the second boundary condition must be one for b . To compute the corresponding ,
T 2 boundary ux f (qb ) = b ub ; b ub + pb ; b b , the `0D' bulk density still needs to be de ned. (In 2D and 3D, the bulk density can be computed in a normal 1D and 2D way, respectively.) In 1D, an appropriate `0D' choice is b = w (b ); for b  0; (3.15a)

b = a (b ); for b < 0:

(3.15b)

15

4. Conclusions

3.4.2 Out ow boundary Here, in addition to (3.14a) and (3.14b), the equations

b = 0 ;

(3.16a)

b = 1

(3.16b)

are available. So, the single boundary condition to be imposed must be ub, pb , or some combination of both. The bulk density b is de ned as in the in ow case. 3.4.3 Non-permeable boundary At a non-permeable boundary (at least) ub = 0 must be imposed, which, given (3.14), already determines pb . Considering a non-permeable boundary as the limit case of an in ow boundary, b must still be imposed. Considering it as the limit of out ow, b follows from the interior solution (b = 1 for left boundary and b = 0 for right). The latter limit case is to be preferred for, e.g., ship hydrodynamics applications. As opposed to in the rst limit case, it allows the water line to freely move up and down the ship hull (Figure 14). Also here, the bulk density may be de ned according to (3.15). fixed

fixed free

s a. Fixed waterline, φb=φ(s)

free

a. Free waterline, φb=φ0 or φb=φ1

Figure 14: Cross section of ship hull with free surface. 4. Conclusions

To accurately compute compressible, immiscible two- uid ows with very large density dierences (water-air ows, e.g.) we have proposed a method that uses a level-set technique to distinguish between the two uids. The resulting equations have been discretized through a nite-volume method. To compute the uxes across the nite-volume walls (the level-set ux being one of the ux-vector components), we have proposed a two- uid, linearized Godunov scheme. The scheme allows a physically correct capturing of the interface across a single cell face, as well as a neat boundary-condition treatment (no sticking of free surfaces to solid walls, e.g.). The scheme combines good physical properties with great simplicity and eciency.

16

References

References 1. B.D. Nichols and C.W. Hirt, Calculating three-dimensional free surface ows in the vicinity

2. 3. 4. 5. 6. 7.

of submerged and exposed structures, J. Comput. Phys., 12, 234{246 (1973). J. Farmer, L. Martinelli and A. Jameson, A fast multigrid method for solving the nonlinear ship wave problem with a free surface, in Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics, Iowa (1993). H.C. Raven, A Solution Method for the Nonlinear Ship Wave Resistance Problem, PhD-thesis, Delft University of Technology (1996). B. Alessandrini and G. Delhommeau, A multigrid velocity-pressure free-surface-elevation fully coupled solver for calculation of turbulent incompressible ow around a hull, in Proceedings of the 21-st Symposium on Naval Hydrodynamics, Trondheim (1996). M.J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge (1978). E.H. van Brummelen, Numerical solution of steady free-surface Navier-Stokes ow, Report MAS-R0018, CWI, Amsterdam (2000). http://www.cwi.nl/ftp/CWIreports/MAS/MASR0018.ps.Z. E.H. van Brummelen, H.C. Raven and B. Koren, Numerical solution of steady free-surface Navier-Stokes ow, in Proceedings of the First International Conference on Computational Fluid Dynamics, Kyoto, 2000 (to appear).

8. E.H.

van Brummelen, B. Koren and H.C. Raven, Ecient solution of steady free-surface Navier-Stokes ow, Report MAS-R0103, CWI,

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

Amsterdam (2001). http://www.cwi.nl/ftp/CWIreports/MAS/MAS-R0103.ps.Z. (to appear in J. Comput. Phys.). S. Piperno, C. Farhat and B. Larrouturou, Partitioned procedures for the transient solution of coupled aeroelastic problems. Part I: model problem, theory and two-dimensional application, Comput. Methods Appl. Mech. Engrg., 124, 79{112 (1995). F.H. Harlow and J.E. Welch, Numerical calculation of time-dependent viscous incompressible

ow of uid with free surface, Physics of Fluids, 8, 2182{2189 (1965). C.W. Hirt and B.D. Nichols, Volume of uid (VOF) method for the dynamics of free boundaries, J. Comput. Phys., 39, 201{225 (1981). J.A. Sethian, Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, Cambridge (1996). W. Mulder, S. Osher and J.A. Sethian, Computing interface motion in compressible gas dynamics, J. Comput. Phys., 100, 209{228 (1992). G.H. Markstein, Chapter B: Theory of ame propagation, in: Non-Steady Flame Propagation (G.H. Markstein, ed.), 5{14, Pergamon Press, New York (1964). S.F. Davis, An interface tracking method for hyperbolic systems of conservation laws, Appl. Numer. Math., 10, 447{472 (1992). J. Zhu and J.A. Sethian, Projection methods coupled to level set interface techniques, J. Comput. Phys., 102, 128{138 (1992). M. Sussman, P. Smereka and S. Osher, A level set approach for computing solutions to incompressible two-phase ow, J. Comput. Phys., 114, 146{159 (1994). H.-K. Zhao, T. Chan, B. Merriman and S. Osher, A variational level set approach to multiphase motion, J. Comput. Phys., 127, 179{195 (1996). Y.C. Chang, T.Y. Hou, B. Merriman and S. Osher, A level set formulation of Eulerian interface capturing methods for incompressible uid ows, J. Comput. Phys., 124, 449{464 (1996).

References

17

20. S. Chen, B. Merriman, S. Osher and P. Smereka, A simple level set method for solving Stefan problems, J. Comput. Phys., 135, 8{29 (1997). 21. J.-P. Cocchi and R. Saurel, A Riemann problem based method for the resolution of compressible multimaterial ows, J. Comput. Phys., 137, 265{298 (1997). 22. T.Y. Hou, Z. Li, S. Osher and H. Zhao, A hybrid method for moving interface problems with application to the Hele-Shaw ow, J. Comput. Phys., 134, 236{252 (1997). 23. B. Koren and A.C.J. Venis, A fed back level-set method for moving material-void interfaces, J. Comput. Appl. Math., 101, 131{152 (1999). 24. D. Adalsteinsson and J.A. Sethian, The fast construction of extension velocities in level set methods, J. Comput. Phys., 148, 2{22 (1999). 25. M. Sussman, A.S. Almgren, J.B. Bell, P. Colella, L.H. Howell and M.L. Welcome, An adaptive level set approach for incompressible two-phase ows, J. Comput. Phys., 148, 81{124 (1999). 26. B.T. Helenbrook, L. Martinelli and C.K. Law, A numerical method for solving incompressible ow problems with a surface of discontinuity, J. Comput. Phys., 148, 366{396 (1999). 27. R.P. Fedkiw, T. Aslam, B. Merriman and S. Osher, A non-oscillatory Eulerian approach to interfaces in multimaterial ows (the ghost uid method), J. Comput. Phys., 152, 457{492 (1999). 28. M. Sussman and E.G. Puckett, A coupled level set and Volume-of-Fluid method for computing 3D and axisymmetric incompressible two-phase ows, J. Comput. Phys., 162, 301{337 (2000). 29. T.D. Aslam, A level-set algorithm for tracking discontinuities in hyperbolic conservation laws I. Scalar equations, J. Comput. Phys., 167, 413{438 (2001). 30. S.J. Osher and G. Tryggvason (eds.), Special issue on computational methods for multiphase

ows, J. Comput. Phys., 169 (2) (2001). 31. C. Aalburg, Experiments in Minimizing Numerical Diusion across a Material Boundary, MSc-thesis, Department of Aerospace Engineering, University of Michigan (1996). Also: http://www.engin.umich.edu/research/cfd/publications/publications.html 32. F.J. Kelecy and R.H. Pletcher, The development of a free surface capturing approach for multidimensional free surface ows in closed containers, J. Comput. Phys., 138, 939{980 (1997). 33. D. Choi and C.L. Merkle, Application of time-iterative schemes to incompressible ow, AIAA J., 23, 1518{1524 (1985). 34. E. Turkel, Preconditioned methods for solving the incompressible and low speed compressible equations, J. Comput. Phys., 72, 277{298 (1987). 35. B. van Leer, W.-T. Lee and P.L. Roe, Characteristic time-stepping or local preconditioning of the Euler equations, AIAA Paper 91-1552 (1991). 36. G. Volpe, Performance of compressible codes at low Mach numbers, AIAA J., 31, 49{56 (1993). 37. Y.-H. Choi and C.L. Merkle, The application of preconditioning in viscous ows, J. Comput. Phys., 105, 207{223 (1993). 38. E. Turkel, Review of preconditioning methods for uid dynamics, Appl. Numer. Math., 12, 257{284 (1993). 39. E. Turkel, A. Fiterman and B. van Leer, Preconditioning and the limit of the compressible to the incompressible ow equations for nite dierence schemes, in Computing the Future: Advances and Prospects for Computational Aerodynamics (M.M. Hafez and D.A. Caughey, eds.), 215{234 Wiley, Chichester (1994). 40. B. Koren, Condition improvement for point relaxation in multigrid, subsonic Euler ow compu-

18

41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51.

tations, Appl. Numer. Math., 16, 457{469 (1995). B. Koren and B. van Leer, Analysis of preconditioning and multigrid for Euler ows with low-subsonic regions, Advances in Comput. Math., 4, 127{144 (1995). P. Wesseling, Principles of Computational Fluid Dynamics, Springer, Berlin (2000). P.G. Tait, Voyage of H.M.S. Challenger, 2(4), Stationery Oce, London (1889). G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge (1967). D.R. van den Heul, C. Vuik and P. Wesseling, A staggered scheme for hyperbolic conservation laws applied to unsteady sheet cavitation, Computing and Vizualization in Science, 2, 63{68 (1999). S.K. Godunov, Finite dierence method for numerical computation of discontinuous solutions of the equations of uid dynamics (Cornell Aeronautical Laboratory Translation from the Russian), Math. Sbornik, 47, 271{306 (1959). S. Osher and F. Solomon, Upwind dierence schemes for hyperbolic systems of conservation laws, Math. Comput., 38, 339{374 (1982). H. van Brummelen and B. Koren, A Godunov-type scheme for capturing water waves, in Proceedings of Godunov Methods, Theory and Applications, Oxford, 1999 (E.F. Toro, ed.), Kluwer, Dordrecht (to appear). B. van Leer, Towards the ultimate conservative dierence scheme, V. A second-order sequel to Godunov's method, J. Comput. Phys., 32, 101{136 (1979), also J. Comput. Phys., 135, 229{248 (1997). E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, A Practical Introduction, Springer, Berlin (1997). R. Courant and K.O. Friedrichs, Supersonic Flow and Shock Waves, Springer, Berlin (1976).

19

A. Two- uid Godunov scheme for Tait's equation of state, derivation and linearization

A. Two-fluid Godunov scheme for Tait's equation of state, derivation and linearization

A.1 State space, Poisson curves and Hugoniot curves For the present two- uid case, just as for the single- uid case, in physical space, the left and right Riemann states (u0 ; p0 ) and (u1 ; p1 ) are connected to the intermediate state (u 21 ; p 21 ) through either a rarefaction wave or a shock wave. The four dierent possible pairs of shock and rarefaction waves that exist are sketched in Figure 15. In the single- uid case, the four states (u 12 ; p 21 ) corresponding with t

t

rarefaction wave

shock wave

( u1/2 , p1/2)

( u1/2 , p1/2)

( u 0 , p 0)

( u 1 , p 1)

x

( u 1 , p 1)

( u 0 , p 0)

a.

b.

t

t

( u1/2 , p1/2)

( u1/2 , p1/2)

( u1 , p 1)

( u 0 , p 0)

c.

( u 0 , p 0)

x

( u 1 , p 1)

x

x

d.

Figure 15: Wave combinations in physical space. those in Figure 15 are determined as the intersection in the (u; p)-state space of either two Poisson curves (Figure 16a), a Poisson and a Hugoniot curve (Figures 16b and 16c), or two Hugoniot curves (Figure 16d). The Poisson curve through a point (uk ; pk ); k = 0; 1 represents all states (u; p) that can be reached from (uk ; pk ) through a rarefaction wave, the Hugoniot curve all points reachable through a shock wave. For details about this (single- uid) theory, see [51], Sections 80 and 81. The speci c forms of the Poisson and Hugoniot curves depend on the equation of state considered. In general, the curves are nonlinear. For a brief description of the two- uid case on the basis of Tait's equation of state, see the next section. A.2 Families of Poisson curves For a two- uid model in which the bulk density description (3.1b) is used, through a point (uk ; pk ); k = 0; 1 in state space, instead of a single Poisson curve, a family of Poisson curves exists, one curve for each value of the Volume-of-Fluid fraction k ; k 2 [0; 1]; k = 0; 1. For rarefaction waves connecting (u0 ; p0 ) to (u 21 ; p 21 ), the family is determined by Zp1 Z p0 1 u+ dp = u + dp; (A.1a)

c

0

c with the bulk density  = (0 ; p) according to (3.1b) and the corresponding speed of sound c = c(0 ; p) according to (2.10), 0 2 [0; 1]. (Equation (A.1a) is equal to the rst equation in (3.5a) with

20 : Hugoniot curve : Poisson curve p

p

0

1

1

1/2 0

1/2 u

u

a.

b.

p

p

1/2

0

1/2

1

0

1 u

u

c.

d.

Figure 16: Poisson and Hugoniot curve combinations in state space, single- uid, combinations a{d correspond with wave pairs in Figures 15a{d. integration to p instead of to .) For rarefactions connecting (u1 ; p1 ) to (u 21 ; p 12 ), the family of Poisson curves is determined by Zp Z p1 1 u , 1 dp = u , dp; (A.1b)

c

1

c with the similar expressions for (1 ; p) and c(1 ; p); 1 2 [0; 1]. Finding formulae for the Poisson curves for general 0 and 1 , (0 ; 1 ) 2 f[0; 1]  [0; 1]g, is tedious. For convenience, here we consider the case 0 = 1 = 1 only (100% water in both the left and right control volume). For  = 1, with Tait's equation of state, it holds

 p + Bw p1  1w

(p) = (1 + B )p w 1

(1 )w ;

 p + Bw p1  2w ,w1

c(p) = (1 + B )p w 1

(A.2a)

s

(c1 )w ; (c1 )w = w (1 +(Bw) )p1 : 1w

(A.2b)

Substitution of (A.2a) and (A.2b) into (A.1a) and (A.1b), and integration, yields for the exact Poisson curves

"  2w ,w1  p0 + Bw p1  2w ,w1 # p + B , 2 w p1 u , u0 = , 1 (1 + B )p , (1 + B )p (c1 )w ; for 0 = 1; w w 1 w 1

(A.3a)

21

A. Two- uid Godunov scheme for Tait's equation of state, derivation and linearization

2

u , u1 = , 1 w

"

p + Bw p1 (1 + Bw )p1

 2w ,w1  p + B p  2w ,w1 # , 1 w1 (c1 )w ; (1 + Bw )p1

for 1 = 1:

(A.3b)

Linearization of (A.3a) and (A.3b) around (u0 ; p0 ) and (u1 ; p1 ), respectively, gives, denoting c by C :

  p , p0 = du ,1 = ,(C ) ; for  = 1; 0w 0 u , u0 dp p0

(A.4a)

  p , p1 = du ,1 = (C ) ; for  = 1; 1w 1 u , u1 dp p1

(A.4b)

which is in agreement with the direct linearization of (3.5a) and (3.7a), which gives (3.10a) and (3.10b), respectively, which { in fact { already are the two families of linearized Poisson curves, with bulk densities 0 and 1 and speeds of sound c0 and c1 valid for all (0 ; 1 ) 2 f[0; 1]  [0; 1]g. A.3 Families of Hugoniot curves In a Lagrangean formulation in which the shock wave is set still, across a shock connecting the preshock state (u0 ; p0 ) and the post-shock state (u 21 ; p 21 ), the ow is always from left to right (Figure 17a). Vice versa, across a shock connecting the pre-shock state (u1 ; p1 ) and the post-shock state (u 21 ; p 12 ), the ow is always from right to left (Figure 17b). u1/2

u0

u1/2

u1

x a. Shock wave connecting ( u 0 , p 0) and ( u1/2 , p1/2 )

x b. Shock wave connecting ( u 1 , p 1) and ( u1/2 , p1/2)

Figure 17: Shock waves in shock frame. In the shock frame, the jump conditions corresponding with the situation of Figures 17a and 17b are, respectively

u



u

 ,1 

1



p = m ,u ; 

(A.5a)

 (A.5b) p =m u ; where m is the mass ow juj through the shock wave. For a derivation of these shock relations in a

Langrangean formulation, see [51], Section 62. From both (A.5a) and (A.5b) it follows

s

] : m = ,[1[p= ]

(A.6)

Using (A.6), the above two jump conditions for momentum can be written as, omitting the subscripts for the post-shock quantities,

s

p , p0 = , p1 , p01 (u , u0 ); 0

,

(A.7a)

22

p , p1 =

s

p , p1 (u , u ): 1 1 1

1

(A.7b)

,

With in equation (A.7a) the bulk densities 0 = 0 (0 ; p0 ) and  = (0 ; p) according to (3.1b), (A.7a) determines one family of Hugoniot curves, one curve for each value of 0 2 [0; 1]. Likewise, (A.7b) does for 1 2 [0; 1]. Equations (A.7a) and (A.7b) are the shock analogies of the rarefaction equations (A.1a) and (A.1b). Working out (A.7a) and (A.7b) for general 0 and 1 on the basis of Tait's equation of state is tedious. As far as the nonlinear equations are concerned, here we also restrict ourselves to the speci c case 0 = 1 = 1 (100% water in the left and right control volume). Elimination of the densities from (A.7a) and (A.7b) with Tait's equation of state yields after some rewriting

v u u1 p , p0 = ,(C0 )w u t w

v u u1 p , p1 = (C1 )w u t w

p+Bw p1 p0 +Bw p1 , 1 1 1 , pp0++BBwwpp11 w



 (u , u0);

p+Bw p1 p1 +Bw p1 , 1 1 1 , pp1++BBwwpp11 w



 (u , u1);

for 0 = 1;

for 1 = 1:

(A.8a)

(A.8b)

Linearization of (A.8a) and (A.8b) around (u0 ; p0) and (u1 ; p1 ), respectively, yields

  p , p0 = du ,1 = ,(C ) ; for  = 1; 0w 0 u , u0 dp p0

(A.9a)

  p , p1 = du ,1 = (C ) ; for  = 1; 1w 1 u , u1 dp p1

(A.9b)

which is identical to (A.4a) and (A.4b) (as it should because the Hugoniot curve and the Poisson curve through a point in state space are tangent to each other in that point). Equations (A.7a) and (A.7b), which are valid for general 0 and 1 , respectively, can also be linearized more directly. For that purpose, rewrite both equations as

rp , p p p , p0 = , 0  , 0 (u , u0 ); 0 = 0 (0 ; p0 );  = (0 ; p); 0 2 [0; 1]; 0

(A.10a)

rp , p p p , p1 = 1  , 1 (u , u1); 1 = 1 (1 ; p1 );  = (1 ; p); 1 2 [0; 1]: (A.10b) 1 q q Linearizing p,p0 as c0 , p,p1 as c1 , and taking p0 and p1 as 0 and 1 , respectively, more ,0

,1

directly gives the linear relations

p , p0 = ,C ; C = C ( ; p );  2 [0; 1]; 0 0 0 0 0 0 u , u0

(A.11a)

p , p1 u , u1 = C1 ; C1 = C1 (1 ; p1 ); 1 2 [0; 1]:

(A.11b)

23

A.4 Intermediate states For the wave pairs sketched in Figure 15, the intermediate states (u 21 ; p 12 ) lie in or at the boundaries of the shaded quadrilaterals in Figures 18a{d. The speci c state (u 12 ; p 21 ) is determined by the speci c values 0 and 1 . Note that due to the convexity of the Poisson curves, the linearized expressions are more sensitive to cavitation, in case of a double rarefaction, than the nonlinear expressions are. A double rarefaction in pure water is even more sensitive than a double rarefaction in pure air; compare (u 21 ; p 12 )0 =1;1 =1 to (u 21 ; p 21 )0 =0;1 =0 in Figure 18a. : linearized Hugoniot curve : linearized Poisson curve p

p 0

α 0 =1

1

1

α 0 =0 α 0 =0

α1 =0 α 0 =1

α1 =1

α1 =0 α1 =1

u

a.

u

b.

p

p 0

0

α1 =1

α1 =0

α 0 =1

α1 =1

α 0 =0

α1 =0

α 0 =0 1

α 0 =1

c.

0

1 u

u

d.

Figure 18: Families of linearized Poisson and Hugoniot curves in state space, two- uid combinations a{d correspond with wave pairs in Figures 15a{d.

24 Table of Contents

Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1.1 Application area: water-air ows : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1.2 Treatment of free-surface ows : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1.2.1 Basic grid techniques : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1.2.2 Moving-grid techniques : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1.2.3 Fixed-grid techniques : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1.3 Point of departure : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1.4 Outline of paper : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 Flow model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2.1 Conservation equations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2.2 level-set equation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2.3 Equations of state : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 Discretization : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3.1 Finite volumes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3.2 Riemann-problem approach : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3.3 Two- uid, linearized Godunov scheme : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3.4 Boundary-condition treatment : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3.4.1 In ow boundary : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3.4.2 Out ow boundary : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3.4.3 Non-permeable boundary : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 Conclusions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : References : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : A Two- uid Godunov scheme for Tait's equation of state, derivation and linearization : : : A.1 State space, Poisson curves and Hugoniot curves : : : : : : : : : : : : : : : : : : : : : : : : : A.2 Families of Poisson curves : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : A.3 Families of Hugoniot curves : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : A.4 Intermediate states : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

1

1 1 1 1 2 3 5 5 6 6 7 8 9 9 10 12 14 14 15 15 15 16 19 19 19 21 23